PlanetPhysics/Uc Locally Compact Quantum Groupoids

Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider locally compact quantum groupoids ($LCQGn$) defined as \htmladdnormallink{locally compact groupoids {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} endowed with a Haar system}, $\nu$, Failed to parse (unknown function "\G"): {\displaystyle (\G,\nu):= ([\grp, G_2, \mu], \nu)} , or as derived from a (\htmladdnormallink{non-commutative {http://planetphysics.us/encyclopedia/AbelianCategory3.html}) weak Hopf algebra} (WHA), with the additional condition of uniform continuity over Failed to parse (unknown function "\grp"): {\displaystyle \grp} defined as follows . Let us also consider a space Failed to parse (unknown function "\grp"): {\displaystyle LUC(\grp)} of left uniformly continuous elements in Failed to parse (unknown function "\grp"): {\displaystyle L^{\infty}(\grp)} defined over $G_{2}$, which is endowed with the induced product topology from the subset $G^{2}$ of composable pairs in the topological groupoid Failed to parse (unknown function "\grp"): {\displaystyle \grp} . This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from `quantum groupoids' derived from a weak Hopf algebra.

C*-algebra Comparison and Example

Consider $LCG$ to be a locally compact quantum group. Then consider the space $LUC(G)$ of left uniformly continuous elements in $L^{\infty }(G)$ introduced in ref. . (The definition according to V. Runde (\em loc. cit.) covers both the space of left uniformly continuous functions on a locally compact group and (Granirer's) uniformly continuous functionals on the Fourier algebra.) Also consider $LUC(G)$ which is then an \htmladdnormallink{operator {http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} system containing the C*-algebra $C_{o}(G)$}. One may compare the groupoid C*-convolution algebra, $G_{CA}$ -- obtained in the general case-- with the C*-algebra $C_{o}(G)$ obtained from $LUC(G)$ in the particular case of uniform continuity over a locally compact group.